In this paper, we present several probabilistic transforms related to classical urn models. These transforms render the dependent random variables describing the urn occupancies into independent random variables with appropriate distributions. This simplifies the analysis of a large number of problems for which a function under investigation depends on the urn occupancies. The approach used for constructing the transforms involves generating functions of combinatorial numbers. We also show, by using Tauberian theorems derived in this paper, that under certain conditions the asymptotic expressions of target functions in the transform domain are identical to the asymptotic expressions in the inverse–transform domain. Therefore, asymptotic information about certain statistics can be gained without evaluating the inverse transform. 1

The general coalescent process with simultaneous multiple mergers of ancestral lines was initially characterized in [13] in terms of a sequence of measures de ned on the nite-dimensional simplices. A more compact characterization of the general coalescent requiring a single probability measure de ned on the in nite simplex was suggested in [17].

Let Y = (y1,y2,...), y1 ≥ y2 ≥ · · · , be the list of sizes of the cycles in the composition of cn transpositions on the set {1,2,...,n}. We prove that if c> 1/2 is constant and n → ∞, the distribution of f(c)Y/n converges to PD(1), the Poisson-Dirichlet distribution with paramenter 1, where the function f is known explicitly. A new proof is presented of the theorem by Diaconis, Mayer-Wolf, Zeitouni and Zerner stating that the PD(1) measure is the unique invariant measure for the uniform coagulation-fragmentation process. 1

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